Intro to commutative property of multiplication

CCSS Math: 3.OA.B.5
Practice changing the order of factors in a multiplication problem and see how it affects the product.

Comparing totals

This array shows 2\greenD{2} rows of dots with 4\purpleD{4} dots in each row. We can use the expression 2×4=8 \greenD{2} \times \purpleD{4}= \goldD{8} to represent the array.
This array shows 4\purpleD{4} rows of dots with 2 \greenD{2} dots in each row. We can use the expression 4×2=8\purpleD{4} \times \greenD{2} = \goldD{8} to represent the array.
In both examples we get a total of 8\goldD{8} dots.
4×2=8\greenD{4} \times\purpleD{2} = \goldD{8} and 2×4=8\purpleD{2} \times \greenD{4} = \goldD{8}
When we change the order of the numbers that we are multiplying the product stays the same.
5×4=20\greenD{5} \times \purpleD{4} = \goldD{20}
4×5=20\purpleD{4}\times \greenD{5} = \goldD{20}
5×4=4×5\greenD{5} \times \purpleD{4} = \purpleD{4}\times \greenD{5}
7×10=70\greenD{7} \times \purpleD{10} = \goldD{70}
10×7=70\purpleD{10}\times \greenD{7} = \goldD{70}
7×10=10×7\greenD{7} \times \purpleD{10} = \purpleD{10}\times \greenD{7}
Practice problem 1a
Match the expressions that are equal to each other.
  • 3×93\times 9
  • 9×49 \times 4
  • 7×47 \times 4
  • 4×94 \times 9
  • 9×39 \times 3
  • 4×74 \times 7

We learned that 4×2=84 \times 2 = 8.
Let's use a picture to decide if 4+24 + 2 =? \stackrel{?}{=} 88.
We can see from the picture that 4+284 + 2 \neq 8. So the answer is not the same when we change the operation.
Practice problem 1b
Which two expressions will give us the same answer?
Choose all answers that apply:
Choose all answers that apply:

Commutative property

The math rule that says the order in which we multiply the factors does not change the product is the commutative property.
Let's use an array to help explain why this works. This array shows 5\goldD{5} rows with 2\blueD{2} dots in each row.
We can find the total number of dots by multiplying the number of rows by the number of dots in each row.
5×2=10\goldD{5} \times \blueD{2} = \greenD{ 10}
If we turned the array on its side we have an array that shows 2\blueD{2} rows with 5\goldD{5} dots in each row.
All we did was tip the array over. The total number of dots did not change.
If we multiply the number of rows by the number of dots in each row we get:
2×5=10\blueD{2} \times \goldD{5} = \greenD{ 10}
The order in which we multiply the numbers 2\blueD{2} and 5\goldD{5} does not matter.
5×2=2×5\goldD{5} \times \blueD{2} = \blueD{2} \times \goldD{5}

Let's try a few problems

This array shows 88 rows with 44 dots in each row.
Problem 2, part A
What would the array look like if we tipped it on its side?
Choose 1 answer:
Choose 1 answer:

Problem 2, part B
88 rows with 44 dots == 44 rows with
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
dots.

Problem 2, part C
8×4=8 \times 4 =
Choose 1 answer:
Choose 1 answer:

Using the commutative property

Describing an array

The commutative property says that the order of the numbers doesn't matter in multiplication.
So the order of the numbers doesn't matter when describing an array.
We can use the expression 5×35 \times 3 to show 55 groups of 33.
Or the expression 3×53 \times 5 to show 33 groups of 55.
Both expressions equal 1515.

Another problem

Practice problem 3
Which two expressions can be used to represent the array?
Choose all answers that apply:
Choose all answers that apply:

Why is the commutative property helpful?

The commutative property can make multiplying more than two numbers easier.
Let's look at an example:
We can multiply 7×2×57 \times 2 \times 5 in two steps:
7×2=147 \times 2 = 14
14×5=7014 \times 5 = 70
We got the right answer, but 14×514 \times 5 is a little tricky to multiply!
Remember that the commutative property lets us change the order of the numbers without changing the answer.
We can switch the 77 and 55 and change the problem to 5×2×75 \times 2 \times 7. Let's see how this makes it easier to multiply:
5×2=105 \times 2 = 10
10×7=7010 \times 7 = 70
Multiplying by 1010 in the second step made it easier to find the product.
Practice problem 4A
Which expressions are the same as 4×3×54 \times 3 \times 5?
Choose all answers that apply:
Choose all answers that apply:

Practice problem 4B
Use the commutative property to rearrange the numbers and solve.
5×3×6=5 \times 3 \times 6 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
Try to rearrange the numbers so you are multiplying by a multiple of 1010.